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2. Analiticheskaya teoriya politropnyh sharov (teoriya Leina-Rittera-Emdena)

Razdely

2.1 Uravnenie Emdena

V etoi glave my budem izuchat' ravnovesnye konfiguracii zvezd, podchinyayushihsya stepennomu (politropnomu) uravneniyu sostoyaniya

$\displaystyle P=K\rho^\gamma=K\rho^{1+{1\over n}},$

gde $\gamma$ -- pokazatel',

-- indeks politropy. Interes k etomu uravneniyu sostoyaniya voznik eshe v proshlom veke, kogda dumali, chto vse zvezdy polnost'yu konvektivny. Pri etom mozhno predpolozhit', chto entropiya postoyanna. Integriruya termodinamicheskie ravenstva

$\displaystyle dE=-Pdv=K\rho^{1+{1\over n}} {1\over \rho^2}d\rho=K\rho^{-1+{1\over n}}d\rho,$

poluchaem vyrazhenie dlya vnutrennei energii

$\displaystyle E=nK\rho^{1\over n}=n{P\over \rho}.$

Otsyuda ental'piya

$\displaystyle H=E+Pv=E+{P\over \rho}=(n+1)\;{P\over \rho}.$

V sluchae ideal'nogo gaza izvestno, chto (

-- teploemkost', ${\cal{R}}$ -- gazovaya postoyannaya)

$\displaystyle E={c_v\over \mu}T, \quad P=\rho {{\cal{R}}T\over \mu}$ ili $\displaystyle \quad {P\over \rho}= {{\cal{R}}T\over \mu}, \quad E={c_v\over {\cal{R}}} {P\over \rho}.$

Dlya odnoatomnogo gaza

$\displaystyle c_v={3\over 2}{\cal{R}}$ i $\displaystyle \quad n={3\over 2}.$

To zhe mozhno vychislit' i dlya mnogoatomnyh gazov, no etot sluchai neinteresen: seichas my znaem o zvezdah neskol'ko bol'she, chem 100 let nazad.

Vvedem peremennuyu $\Theta$ takim obrazom, chtoby

$\displaystyle \rho=\lambda \Theta^n, \quad P=K \lambda^{1+{1\over n}} \Theta^{n+1},$

t.e. imeem $\displaystyle 𣏵\rho\over \rho_c}={\lambda \Theta^n\over \rho_c}, \quad {P\over \rho}=K\lambda^{1\over n}\Theta$ i $\displaystyle \quad H=(n+1)K\lambda^{1\over n} \Theta.$

Dlya ideal'nogo gaza s postoyannoi teploemkost'yu velichina $\Theta$ proporcional'na temperature. Voz'mem uslovie ravnovesiya v vide

$\displaystyle \varphi+H=$ const $\displaystyle$

i podeistvuem na nego operatorom $\Delta$ . Laplasian $\varphi$ est' $4\pi G \rho$ , t.e. $\Delta \varphi=4\pi G \lambda \Theta^n$ , a laplasian

est' $(n+1)K\lambda^{1/n}\Delta \Theta$ , i uravnenie ravnovesiya zapishetsya v vide

$\displaystyle 4\pi G \lambda \Theta^n+(n+1)K\lambda^{1/n}\Delta \Theta=0 .$

Budem uproshat' poluchennoe sootnoshenie, izmenyaya masshtab, t.e. vvodya peremennuyu $\xi$ cherez sootnoshenie $r=\alpha \xi$ ,

$\displaystyle 4\pi G \lambda \Theta^n+{(n+1)K\lambda^{1/n}\over \alpha^2} \Delta_\xi \Theta=0,$

gde $\Delta_\xi={1\over \xi^2} {\partial\over \partial \xi} \xi^2 {\partial\over \partial \xi}$ .

Vyberem $\alpha$ tak, chtoby $4\pi G \lambda=(n+1)K\lambda^{1/n}/\alpha^2$ . Togda uravnenie ravnovesiya zapishetsya v vide

$\displaystyle \Delta_\xi \Theta+\Theta^n=0 ,$ ili $\displaystyle \quad{1\over \xi^2} {d\over d \xi} \xi^2 {d \Theta\over d \xi}+\Theta^n=0.$

$\begin{wrapfigure}{r}{0.5\textwidth} \epsfxsize =0.45\textwidth \hbox to0.5\textwidth{\hss\epsfbox{fig/f12.ai}\hss} \end{wrapfigure}$

Ris. 12.

Takim obrazom, pri dannom uravnenie ravnovesiya odno i to zhe dlya zvezd lyuboi massy. Reshaya uravnenie pri granichnyh usloviyah $\Theta(0)=1, \;\left.{d \Theta \over d \xi}\right\vert _{\xi=0}=0$ (t.e. polozhiv $\lambda=\rho_c$ ), poluchim monotonnoe ubyvanie $\Theta$ ot edinicy k nulyu (ris. 12). Znachenie $\xi_1$ , gde $\Theta(\xi_1) =0$ , yavlyaetsya granicei zvezdy. Plotnost' $\rho$ , proporcional'naya $\Theta^n$ , pri spadaet bolee kruto, chem $\Theta$ . My uzhe pokazyvali v razdele 1.5, chto pri stepennom uravnenii sostoyaniya na krayu zvezdy $\rho \sim (R-r)^{1/(\gamma-1)}$ . No $\gamma=1+1/n$ , t.e. $\rho \sim (R-r)^n \sim \Theta^n$ . Poetomu $\Theta \sim (\xi_1-\xi)$ i vblizi $\xi=\xi_1$ velichina $\Theta$ prohodit nul' s konechnoi proizvodnoi, hotya $\Theta^n$ ``steletsya'' (pri ), t.e. podhodit k nulyu, kasayas' osi absciss.

Yasno, chto dlya zvezd s razlichnymi $\rho_c$ i krivye s odinakovymi podobny. Dostatochno znat' tol'ko odnu funkciyu $\Theta(\xi)$ . Podcherknem vazhnost' granichnogo usloviya $\left.{d \Theta\over d \xi}\right\vert _{\xi=0}=0$ . Obratnoe $\left(\left.{d \Theta\over d \xi}\right\vert _{\xi=0} \ne 0\right)$ oznachalo by konechnyi skachok uskoreniya v centre (t.e. osobennost')^2.1.

Neskol'ko avtorov v proshlom veke chislenno prointegrirovali uravnenie dlya razlichnyh . V chastnosti, Emden poluchil tablicy $\Theta_n(\xi)$ c bol'shoi tochnost'yu. Znachenie etih vychislenii teper' neveliko, tak kak raschet real'nyh zvezd provoditsya s uchetom fizicheskih faktorov, sovershenno ne uchityvaemyh v politropnoi teorii (nestepennoe uravnenie sostoyaniya; istinnaya svyaz' i $\rho$ poluchaetsya iz rassmotreniya vseh processov, vklyuchaya perenos izlucheniya, yadernye reakcii). Odnako dlya kachestvennyh issledovanii reshenie uravnenii Emdena ves'ma polezno. Naprimer, s pomosh'yu politropnoi modeli legko pokazat' nevozmozhnost' sushestvovaniya sverhmassivnyh zvezd. Eto vazhno dlya problemy kvazarov.

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